// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_EIGENSOLVER_H
#define EIGEN_EIGENSOLVER_H

#include "./RealSchur.h"

namespace Eigen {

/** \eigenvalues_module \ingroup Eigenvalues_Module
 *
 *
 * \class EigenSolver
 *
 * \brief Computes eigenvalues and eigenvectors of general matrices
 *
 * \tparam _MatrixType the type of the matrix of which we are computing the
 * eigendecomposition; this is expected to be an instantiation of the Matrix
 * class template. Currently, only real matrices are supported.
 *
 * The eigenvalues and eigenvectors of a matrix \f$ A \f$ are scalars
 * \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda v \f$.  If
 * \f$ D \f$ is a diagonal matrix with the eigenvalues on the diagonal, and
 * \f$ V \f$ is a matrix with the eigenvectors as its columns, then \f$ A V =
 * V D \f$. The matrix \f$ V \f$ is almost always invertible, in which case we
 * have \f$ A = V D V^{-1} \f$. This is called the eigendecomposition.
 *
 * The eigenvalues and eigenvectors of a matrix may be complex, even when the
 * matrix is real. However, we can choose real matrices \f$ V \f$ and \f$ D
 * \f$ satisfying \f$ A V = V D \f$, just like the eigendecomposition, if the
 * matrix \f$ D \f$ is not required to be diagonal, but if it is allowed to
 * have blocks of the form
 * \f[ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f]
 * (where \f$ u \f$ and \f$ v \f$ are real numbers) on the diagonal.  These
 * blocks correspond to complex eigenvalue pairs \f$ u \pm iv \f$. We call
 * this variant of the eigendecomposition the pseudo-eigendecomposition.
 *
 * Call the function compute() to compute the eigenvalues and eigenvectors of
 * a given matrix. Alternatively, you can use the
 * EigenSolver(const MatrixType&, bool) constructor which computes the
 * eigenvalues and eigenvectors at construction time. Once the eigenvalue and
 * eigenvectors are computed, they can be retrieved with the eigenvalues() and
 * eigenvectors() functions. The pseudoEigenvalueMatrix() and
 * pseudoEigenvectors() methods allow the construction of the
 * pseudo-eigendecomposition.
 *
 * The documentation for EigenSolver(const MatrixType&, bool) contains an
 * example of the typical use of this class.
 *
 * \note The implementation is adapted from
 * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
 * Their code is based on EISPACK.
 *
 * \sa MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver
 */
template<typename _MatrixType>
class EigenSolver
{
  public:
	/** \brief Synonym for the template parameter \p _MatrixType. */
	typedef _MatrixType MatrixType;

	enum
	{
		RowsAtCompileTime = MatrixType::RowsAtCompileTime,
		ColsAtCompileTime = MatrixType::ColsAtCompileTime,
		Options = MatrixType::Options,
		MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
		MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
	};

	/** \brief Scalar type for matrices of type #MatrixType. */
	typedef typename MatrixType::Scalar Scalar;
	typedef typename NumTraits<Scalar>::Real RealScalar;
	typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3

	/** \brief Complex scalar type for #MatrixType.
	 *
	 * This is \c std::complex<Scalar> if #Scalar is real (e.g.,
	 * \c float or \c double) and just \c Scalar if #Scalar is
	 * complex.
	 */
	typedef std::complex<RealScalar> ComplexScalar;

	/** \brief Type for vector of eigenvalues as returned by eigenvalues().
	 *
	 * This is a column vector with entries of type #ComplexScalar.
	 * The length of the vector is the size of #MatrixType.
	 */
	typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;

	/** \brief Type for matrix of eigenvectors as returned by eigenvectors().
	 *
	 * This is a square matrix with entries of type #ComplexScalar.
	 * The size is the same as the size of #MatrixType.
	 */
	typedef Matrix<ComplexScalar,
				   RowsAtCompileTime,
				   ColsAtCompileTime,
				   Options,
				   MaxRowsAtCompileTime,
				   MaxColsAtCompileTime>
		EigenvectorsType;

	/** \brief Default constructor.
	 *
	 * The default constructor is useful in cases in which the user intends to
	 * perform decompositions via EigenSolver::compute(const MatrixType&, bool).
	 *
	 * \sa compute() for an example.
	 */
	EigenSolver()
		: m_eivec()
		, m_eivalues()
		, m_isInitialized(false)
		, m_eigenvectorsOk(false)
		, m_realSchur()
		, m_matT()
		, m_tmp()
	{
	}

	/** \brief Default constructor with memory preallocation
	 *
	 * Like the default constructor but with preallocation of the internal data
	 * according to the specified problem \a size.
	 * \sa EigenSolver()
	 */
	explicit EigenSolver(Index size)
		: m_eivec(size, size)
		, m_eivalues(size)
		, m_isInitialized(false)
		, m_eigenvectorsOk(false)
		, m_realSchur(size)
		, m_matT(size, size)
		, m_tmp(size)
	{
	}

	/** \brief Constructor; computes eigendecomposition of given matrix.
	 *
	 * \param[in]  matrix  Square matrix whose eigendecomposition is to be computed.
	 * \param[in]  computeEigenvectors  If true, both the eigenvectors and the
	 *    eigenvalues are computed; if false, only the eigenvalues are
	 *    computed.
	 *
	 * This constructor calls compute() to compute the eigenvalues
	 * and eigenvectors.
	 *
	 * Example: \include EigenSolver_EigenSolver_MatrixType.cpp
	 * Output: \verbinclude EigenSolver_EigenSolver_MatrixType.out
	 *
	 * \sa compute()
	 */
	template<typename InputType>
	explicit EigenSolver(const EigenBase<InputType>& matrix, bool computeEigenvectors = true)
		: m_eivec(matrix.rows(), matrix.cols())
		, m_eivalues(matrix.cols())
		, m_isInitialized(false)
		, m_eigenvectorsOk(false)
		, m_realSchur(matrix.cols())
		, m_matT(matrix.rows(), matrix.cols())
		, m_tmp(matrix.cols())
	{
		compute(matrix.derived(), computeEigenvectors);
	}

	/** \brief Returns the eigenvectors of given matrix.
	 *
	 * \returns  %Matrix whose columns are the (possibly complex) eigenvectors.
	 *
	 * \pre Either the constructor
	 * EigenSolver(const MatrixType&,bool) or the member function
	 * compute(const MatrixType&, bool) has been called before, and
	 * \p computeEigenvectors was set to true (the default).
	 *
	 * Column \f$ k \f$ of the returned matrix is an eigenvector corresponding
	 * to eigenvalue number \f$ k \f$ as returned by eigenvalues().  The
	 * eigenvectors are normalized to have (Euclidean) norm equal to one. The
	 * matrix returned by this function is the matrix \f$ V \f$ in the
	 * eigendecomposition \f$ A = V D V^{-1} \f$, if it exists.
	 *
	 * Example: \include EigenSolver_eigenvectors.cpp
	 * Output: \verbinclude EigenSolver_eigenvectors.out
	 *
	 * \sa eigenvalues(), pseudoEigenvectors()
	 */
	EigenvectorsType eigenvectors() const;

	/** \brief Returns the pseudo-eigenvectors of given matrix.
	 *
	 * \returns  Const reference to matrix whose columns are the pseudo-eigenvectors.
	 *
	 * \pre Either the constructor
	 * EigenSolver(const MatrixType&,bool) or the member function
	 * compute(const MatrixType&, bool) has been called before, and
	 * \p computeEigenvectors was set to true (the default).
	 *
	 * The real matrix \f$ V \f$ returned by this function and the
	 * block-diagonal matrix \f$ D \f$ returned by pseudoEigenvalueMatrix()
	 * satisfy \f$ AV = VD \f$.
	 *
	 * Example: \include EigenSolver_pseudoEigenvectors.cpp
	 * Output: \verbinclude EigenSolver_pseudoEigenvectors.out
	 *
	 * \sa pseudoEigenvalueMatrix(), eigenvectors()
	 */
	const MatrixType& pseudoEigenvectors() const
	{
		eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
		eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
		return m_eivec;
	}

	/** \brief Returns the block-diagonal matrix in the pseudo-eigendecomposition.
	 *
	 * \returns  A block-diagonal matrix.
	 *
	 * \pre Either the constructor
	 * EigenSolver(const MatrixType&,bool) or the member function
	 * compute(const MatrixType&, bool) has been called before.
	 *
	 * The matrix \f$ D \f$ returned by this function is real and
	 * block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2
	 * blocks of the form
	 * \f$ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f$.
	 * These blocks are not sorted in any particular order.
	 * The matrix \f$ D \f$ and the matrix \f$ V \f$ returned by
	 * pseudoEigenvectors() satisfy \f$ AV = VD \f$.
	 *
	 * \sa pseudoEigenvectors() for an example, eigenvalues()
	 */
	MatrixType pseudoEigenvalueMatrix() const;

	/** \brief Returns the eigenvalues of given matrix.
	 *
	 * \returns A const reference to the column vector containing the eigenvalues.
	 *
	 * \pre Either the constructor
	 * EigenSolver(const MatrixType&,bool) or the member function
	 * compute(const MatrixType&, bool) has been called before.
	 *
	 * The eigenvalues are repeated according to their algebraic multiplicity,
	 * so there are as many eigenvalues as rows in the matrix. The eigenvalues
	 * are not sorted in any particular order.
	 *
	 * Example: \include EigenSolver_eigenvalues.cpp
	 * Output: \verbinclude EigenSolver_eigenvalues.out
	 *
	 * \sa eigenvectors(), pseudoEigenvalueMatrix(),
	 *     MatrixBase::eigenvalues()
	 */
	const EigenvalueType& eigenvalues() const
	{
		eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
		return m_eivalues;
	}

	/** \brief Computes eigendecomposition of given matrix.
	 *
	 * \param[in]  matrix  Square matrix whose eigendecomposition is to be computed.
	 * \param[in]  computeEigenvectors  If true, both the eigenvectors and the
	 *    eigenvalues are computed; if false, only the eigenvalues are
	 *    computed.
	 * \returns    Reference to \c *this
	 *
	 * This function computes the eigenvalues of the real matrix \p matrix.
	 * The eigenvalues() function can be used to retrieve them.  If
	 * \p computeEigenvectors is true, then the eigenvectors are also computed
	 * and can be retrieved by calling eigenvectors().
	 *
	 * The matrix is first reduced to real Schur form using the RealSchur
	 * class. The Schur decomposition is then used to compute the eigenvalues
	 * and eigenvectors.
	 *
	 * The cost of the computation is dominated by the cost of the
	 * Schur decomposition, which is very approximately \f$ 25n^3 \f$
	 * (where \f$ n \f$ is the size of the matrix) if \p computeEigenvectors
	 * is true, and \f$ 10n^3 \f$ if \p computeEigenvectors is false.
	 *
	 * This method reuses of the allocated data in the EigenSolver object.
	 *
	 * Example: \include EigenSolver_compute.cpp
	 * Output: \verbinclude EigenSolver_compute.out
	 */
	template<typename InputType>
	EigenSolver& compute(const EigenBase<InputType>& matrix, bool computeEigenvectors = true);

	/** \returns NumericalIssue if the input contains INF or NaN values or overflow occurred. Returns Success otherwise.
	 */
	ComputationInfo info() const
	{
		eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
		return m_info;
	}

	/** \brief Sets the maximum number of iterations allowed. */
	EigenSolver& setMaxIterations(Index maxIters)
	{
		m_realSchur.setMaxIterations(maxIters);
		return *this;
	}

	/** \brief Returns the maximum number of iterations. */
	Index getMaxIterations() { return m_realSchur.getMaxIterations(); }

  private:
	void doComputeEigenvectors();

  protected:
	static void check_template_parameters()
	{
		EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
		EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL);
	}

	MatrixType m_eivec;
	EigenvalueType m_eivalues;
	bool m_isInitialized;
	bool m_eigenvectorsOk;
	ComputationInfo m_info;
	RealSchur<MatrixType> m_realSchur;
	MatrixType m_matT;

	typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
	ColumnVectorType m_tmp;
};

template<typename MatrixType>
MatrixType
EigenSolver<MatrixType>::pseudoEigenvalueMatrix() const
{
	eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
	const RealScalar precision = RealScalar(2) * NumTraits<RealScalar>::epsilon();
	Index n = m_eivalues.rows();
	MatrixType matD = MatrixType::Zero(n, n);
	for (Index i = 0; i < n; ++i) {
		if (internal::isMuchSmallerThan(
				numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i)), precision))
			matD.coeffRef(i, i) = numext::real(m_eivalues.coeff(i));
		else {
			matD.template block<2, 2>(i, i) << numext::real(m_eivalues.coeff(i)), numext::imag(m_eivalues.coeff(i)),
				-numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i));
			++i;
		}
	}
	return matD;
}

template<typename MatrixType>
typename EigenSolver<MatrixType>::EigenvectorsType
EigenSolver<MatrixType>::eigenvectors() const
{
	eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
	eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
	const RealScalar precision = RealScalar(2) * NumTraits<RealScalar>::epsilon();
	Index n = m_eivec.cols();
	EigenvectorsType matV(n, n);
	for (Index j = 0; j < n; ++j) {
		if (internal::isMuchSmallerThan(
				numext::imag(m_eivalues.coeff(j)), numext::real(m_eivalues.coeff(j)), precision) ||
			j + 1 == n) {
			// we have a real eigen value
			matV.col(j) = m_eivec.col(j).template cast<ComplexScalar>();
			matV.col(j).normalize();
		} else {
			// we have a pair of complex eigen values
			for (Index i = 0; i < n; ++i) {
				matV.coeffRef(i, j) = ComplexScalar(m_eivec.coeff(i, j), m_eivec.coeff(i, j + 1));
				matV.coeffRef(i, j + 1) = ComplexScalar(m_eivec.coeff(i, j), -m_eivec.coeff(i, j + 1));
			}
			matV.col(j).normalize();
			matV.col(j + 1).normalize();
			++j;
		}
	}
	return matV;
}

template<typename MatrixType>
template<typename InputType>
EigenSolver<MatrixType>&
EigenSolver<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeEigenvectors)
{
	check_template_parameters();

	using numext::isfinite;
	using std::abs;
	using std::sqrt;
	eigen_assert(matrix.cols() == matrix.rows());

	// Reduce to real Schur form.
	m_realSchur.compute(matrix.derived(), computeEigenvectors);

	m_info = m_realSchur.info();

	if (m_info == Success) {
		m_matT = m_realSchur.matrixT();
		if (computeEigenvectors)
			m_eivec = m_realSchur.matrixU();

		// Compute eigenvalues from matT
		m_eivalues.resize(matrix.cols());
		Index i = 0;
		while (i < matrix.cols()) {
			if (i == matrix.cols() - 1 || m_matT.coeff(i + 1, i) == Scalar(0)) {
				m_eivalues.coeffRef(i) = m_matT.coeff(i, i);
				if (!(isfinite)(m_eivalues.coeffRef(i))) {
					m_isInitialized = true;
					m_eigenvectorsOk = false;
					m_info = NumericalIssue;
					return *this;
				}
				++i;
			} else {
				Scalar p = Scalar(0.5) * (m_matT.coeff(i, i) - m_matT.coeff(i + 1, i + 1));
				Scalar z;
				// Compute z = sqrt(abs(p * p + m_matT.coeff(i+1, i) * m_matT.coeff(i, i+1)));
				// without overflow
				{
					Scalar t0 = m_matT.coeff(i + 1, i);
					Scalar t1 = m_matT.coeff(i, i + 1);
					Scalar maxval = numext::maxi<Scalar>(abs(p), numext::maxi<Scalar>(abs(t0), abs(t1)));
					t0 /= maxval;
					t1 /= maxval;
					Scalar p0 = p / maxval;
					z = maxval * sqrt(abs(p0 * p0 + t0 * t1));
				}

				m_eivalues.coeffRef(i) = ComplexScalar(m_matT.coeff(i + 1, i + 1) + p, z);
				m_eivalues.coeffRef(i + 1) = ComplexScalar(m_matT.coeff(i + 1, i + 1) + p, -z);
				if (!((isfinite)(m_eivalues.coeffRef(i)) && (isfinite)(m_eivalues.coeffRef(i + 1)))) {
					m_isInitialized = true;
					m_eigenvectorsOk = false;
					m_info = NumericalIssue;
					return *this;
				}
				i += 2;
			}
		}

		// Compute eigenvectors.
		if (computeEigenvectors)
			doComputeEigenvectors();
	}

	m_isInitialized = true;
	m_eigenvectorsOk = computeEigenvectors;

	return *this;
}

template<typename MatrixType>
void
EigenSolver<MatrixType>::doComputeEigenvectors()
{
	using std::abs;
	const Index size = m_eivec.cols();
	const Scalar eps = NumTraits<Scalar>::epsilon();

	// inefficient! this is already computed in RealSchur
	Scalar norm(0);
	for (Index j = 0; j < size; ++j) {
		norm += m_matT.row(j).segment((std::max)(j - 1, Index(0)), size - (std::max)(j - 1, Index(0))).cwiseAbs().sum();
	}

	// Backsubstitute to find vectors of upper triangular form
	if (norm == Scalar(0)) {
		return;
	}

	for (Index n = size - 1; n >= 0; n--) {
		Scalar p = m_eivalues.coeff(n).real();
		Scalar q = m_eivalues.coeff(n).imag();

		// Scalar vector
		if (q == Scalar(0)) {
			Scalar lastr(0), lastw(0);
			Index l = n;

			m_matT.coeffRef(n, n) = Scalar(1);
			for (Index i = n - 1; i >= 0; i--) {
				Scalar w = m_matT.coeff(i, i) - p;
				Scalar r = m_matT.row(i).segment(l, n - l + 1).dot(m_matT.col(n).segment(l, n - l + 1));

				if (m_eivalues.coeff(i).imag() < Scalar(0)) {
					lastw = w;
					lastr = r;
				} else {
					l = i;
					if (m_eivalues.coeff(i).imag() == Scalar(0)) {
						if (w != Scalar(0))
							m_matT.coeffRef(i, n) = -r / w;
						else
							m_matT.coeffRef(i, n) = -r / (eps * norm);
					} else // Solve real equations
					{
						Scalar x = m_matT.coeff(i, i + 1);
						Scalar y = m_matT.coeff(i + 1, i);
						Scalar denom = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) +
									   m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag();
						Scalar t = (x * lastr - lastw * r) / denom;
						m_matT.coeffRef(i, n) = t;
						if (abs(x) > abs(lastw))
							m_matT.coeffRef(i + 1, n) = (-r - w * t) / x;
						else
							m_matT.coeffRef(i + 1, n) = (-lastr - y * t) / lastw;
					}

					// Overflow control
					Scalar t = abs(m_matT.coeff(i, n));
					if ((eps * t) * t > Scalar(1))
						m_matT.col(n).tail(size - i) /= t;
				}
			}
		} else if (q < Scalar(0) && n > 0) // Complex vector
		{
			Scalar lastra(0), lastsa(0), lastw(0);
			Index l = n - 1;

			// Last vector component imaginary so matrix is triangular
			if (abs(m_matT.coeff(n, n - 1)) > abs(m_matT.coeff(n - 1, n))) {
				m_matT.coeffRef(n - 1, n - 1) = q / m_matT.coeff(n, n - 1);
				m_matT.coeffRef(n - 1, n) = -(m_matT.coeff(n, n) - p) / m_matT.coeff(n, n - 1);
			} else {
				ComplexScalar cc = ComplexScalar(Scalar(0), -m_matT.coeff(n - 1, n)) /
								   ComplexScalar(m_matT.coeff(n - 1, n - 1) - p, q);
				m_matT.coeffRef(n - 1, n - 1) = numext::real(cc);
				m_matT.coeffRef(n - 1, n) = numext::imag(cc);
			}
			m_matT.coeffRef(n, n - 1) = Scalar(0);
			m_matT.coeffRef(n, n) = Scalar(1);
			for (Index i = n - 2; i >= 0; i--) {
				Scalar ra = m_matT.row(i).segment(l, n - l + 1).dot(m_matT.col(n - 1).segment(l, n - l + 1));
				Scalar sa = m_matT.row(i).segment(l, n - l + 1).dot(m_matT.col(n).segment(l, n - l + 1));
				Scalar w = m_matT.coeff(i, i) - p;

				if (m_eivalues.coeff(i).imag() < Scalar(0)) {
					lastw = w;
					lastra = ra;
					lastsa = sa;
				} else {
					l = i;
					if (m_eivalues.coeff(i).imag() == RealScalar(0)) {
						ComplexScalar cc = ComplexScalar(-ra, -sa) / ComplexScalar(w, q);
						m_matT.coeffRef(i, n - 1) = numext::real(cc);
						m_matT.coeffRef(i, n) = numext::imag(cc);
					} else {
						// Solve complex equations
						Scalar x = m_matT.coeff(i, i + 1);
						Scalar y = m_matT.coeff(i + 1, i);
						Scalar vr = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) +
									m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag() - q * q;
						Scalar vi = (m_eivalues.coeff(i).real() - p) * Scalar(2) * q;
						if ((vr == Scalar(0)) && (vi == Scalar(0)))
							vr = eps * norm * (abs(w) + abs(q) + abs(x) + abs(y) + abs(lastw));

						ComplexScalar cc =
							ComplexScalar(x * lastra - lastw * ra + q * sa, x * lastsa - lastw * sa - q * ra) /
							ComplexScalar(vr, vi);
						m_matT.coeffRef(i, n - 1) = numext::real(cc);
						m_matT.coeffRef(i, n) = numext::imag(cc);
						if (abs(x) > (abs(lastw) + abs(q))) {
							m_matT.coeffRef(i + 1, n - 1) =
								(-ra - w * m_matT.coeff(i, n - 1) + q * m_matT.coeff(i, n)) / x;
							m_matT.coeffRef(i + 1, n) = (-sa - w * m_matT.coeff(i, n) - q * m_matT.coeff(i, n - 1)) / x;
						} else {
							cc = ComplexScalar(-lastra - y * m_matT.coeff(i, n - 1), -lastsa - y * m_matT.coeff(i, n)) /
								 ComplexScalar(lastw, q);
							m_matT.coeffRef(i + 1, n - 1) = numext::real(cc);
							m_matT.coeffRef(i + 1, n) = numext::imag(cc);
						}
					}

					// Overflow control
					Scalar t = numext::maxi<Scalar>(abs(m_matT.coeff(i, n - 1)), abs(m_matT.coeff(i, n)));
					if ((eps * t) * t > Scalar(1))
						m_matT.block(i, n - 1, size - i, 2) /= t;
				}
			}

			// We handled a pair of complex conjugate eigenvalues, so need to skip them both
			n--;
		} else {
			eigen_assert(0 &&
						 "Internal bug in EigenSolver (INF or NaN has not been detected)"); // this should not happen
		}
	}

	// Back transformation to get eigenvectors of original matrix
	for (Index j = size - 1; j >= 0; j--) {
		m_tmp.noalias() = m_eivec.leftCols(j + 1) * m_matT.col(j).segment(0, j + 1);
		m_eivec.col(j) = m_tmp;
	}
}

} // end namespace Eigen

#endif // EIGEN_EIGENSOLVER_H
